Hi together - this question has arisen while I was contemplating a possible solution to another question I have posted (see <a href="http://mathoverflow.net/questions/98917/harmonic-functions-and-itos-formula-with-respect-to-the-generator-of-a-reflectin">Harmonic functions and Itô‘s Formula with respect to the generator of a Reflecting Brownian Motion</a>)

Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost everywhere on the boundary with respect to the surface measure. Furthermore define $X_t$ to be a process which behaves like the standard Brownian Motion in $(0,1)^2$ and has normal reflection on the boundary (the classical reflected brownian motion) 

It is known, that $X_t$ is a Semimartingale and can be expressed as: 
$X_t=X_0+W_t+\frac{1}{2}\int_0^t v(X_s)d L_s$ where $W^t$ is the Standard Brownian Motion, $L$ the boundary local time and $v$ is the inword normal vektor field on the boundary.
Obviosuly $V_t=\frac{1}{2}\int_0^t v(X_s)d L_s$ is thus a process of bounded Variation.

> **Question**:
>would $f(X_t)=f(X_0)+\int_0^t \nabla f(X_s)dW_s+\int_0^t \nabla f(X_s)dV_s+\int_0^t \Delta f(X_s)dt$ 
be still vaild if $X_0=x\in \partial [0,1]^2$ ?

My main concern is that neither $\nabla f$ nor $\Delta f$ are defined on $\partial [0,1]^2$ - **is there a way to modify the Defintion of $f$ so that Itô's formula coul still be used ?** - so basically how does one define Derivatives on the boundary - is there a standard way? (have been studying math for a while and am wondering why this question has never occured to me before)

My first idea would be to extend $f$ to say $[-\epsilon, 1+\epsilon]$ and use the derivative of the extended function at $x\in\partial [0,1]^2$

Thanks in Advance
:)